What type of neural network would be most feasible for playing a realtime game?
I recommend you read up on reinforcement learning. Seeing how AirHockey is similar to the old Atari game Pong, here is a write-up (with code) about how to implement a simple neural network, that plays the game; Deep Reinforcement Learning: Pong from Pixels.

As far as the choice of layers go, again, you should read up on deep reinforcement learning, perhaps study some existing models that work, for a similar task and take their architecture as your starting point. The above link is a nice introduction into such thinking.

As a continuation, I would recomend you employ one of the well known, and developed, libraries, such as TensorFlow or pyTorch. They will allow you to run your models on GPU, like you desire.

For a toolkit, that specializes to reinforcement learning, see openAI gym.